aspects may be gained by imagining how a nest of inter-related cubes
(made of wire, so as to interpenetrate), combined into a single
symmetrical figure of three-dimensional space, would appear
from several different directions. Each view would yield new
space-subdivisions, and all would be rhythmical--susceptible,
therefore, of translation into ornament. C and D represent such
translations of A and B.

In order to fix these unfamiliar ideas more firmly in the reader's
mind, let him submit himself to one more exercise of the creative
imagination, and construct, by a slightly different method, a
representation of a hexadecahedroid, or 16-hedroid, on a plane. This
regular solid of four-dimensional space consists of sixteen cells,
each a regular tetrahedron, thirty-two triangular faces, twenty-four
edges and eight vertices. It is the correlative of the octahedron of
three-dimensional space.

First it is necessary to establish our four axes, all mutually
at right angles. If we draw three lines intersecting at a point,
subtending angles of 60 degrees each, it is not difficult to
conceive of these lines as being at right angles with one another
in three-dimensional space. The fourth axis we will assume to pass
vertically through the point of intersection of the three lines,
so that we see it only in cross-section, that is, as a point. It is
important to remember that all of the angles made by the four axes
are right angles--a thing possible only in a space of four dimensions.
Because the 16-hedroid is a symmetrical hyper-solid all of its
eight apexes will be equidistant from the centre of a containing
hyper-sphere, whose "surface" these will intersect at symmetrically
disposed points. These apexes are established in our representation by